Question

For the following exercises, determine the eccentricity and identify the conic. Sketch the conic. $$ r=\frac{4}{3-2 \cos \theta} $$

Short answer

The eccentricity is \( \frac{2}{3} \) and the conic is an ellipse.

Step-by-step solution

Rewrite the Equation in Standard Polar Form

The given polar equation is \( r = \frac{4}{3 - 2\cos\theta} \). In standard polar form, an equation of a conic section is \( r = \frac{ed}{1 + e \cos \theta} \). Compare this with the given equation to find \( e \) and \( d \).

Identify the Eccentricity

Rewrite the given equation as \( r = \frac{4}{1 - (-\frac{2}{3}) \cos \theta} \) to match the standard form \( r = \frac{ed}{1 - e\cos\theta} \). From this, \( ed = 4 \) and \( e = \frac{2}{3} \). Since \( e = \frac{2}{3} < 1 \), the conic is an ellipse.

Confirm the Conic Type

With \( e = \frac{2}{3} \), we know that \( 0 < e < 1 \), which confirms that the conic is indeed an ellipse. A value of \( e = 1 \) or more would indicate a parabola or hyperbola, respectively.

Sketch the Conic

Plot the conic based on the eccentricity and polar equation. Since it is an ellipse centered at the pole, draw an elongated shape around the pole in the direction of \( \theta \). The maximum distance from the origin is where the denomiantor \( 3 - 2\cos\theta \) is smallest.

Key concepts

Polar Equations of Conic Sections

In the world of conic sections, polar equations are a common way to represent shapes such as circles, ellipses, parabolas, and hyperbolas. A polar equation is expressed in terms of the radius \( r \) and angle \( \theta \), which provide the position of a point in a plane using polar coordinates. In essence, polar coordinates offer an elegant way to describe the curves and forms that can result from slicing a cone at different angles.
The general form of a polar equation for a conic section is \( r = \frac{ed}{1 + e\cos\theta} \), where \( e \) represents eccentricity, and \( d \) is a constant related to the conic's dimensions. The structure of this equation can change slightly depending on the conic and the direction of the angle.

• For ellipses: \( e < 1 \)
• For parabolas: \( e = 1 \)
• For hyperbolas: \( e > 1 \)

The elegance of polar equations lies in their ability to simplify the representation and transformation of complex curves. They can also effortlessly bridge the distance between Cartesian and polar viewpoints.

Understanding Eccentricity

Eccentricity is a fundamental concept in the study of conic sections and helps distinguish one conic from another. It defines how "stretched" a conic section is. In the context of ellipses, eccentricity \( e \) measures the deviation of the ellipse from being a perfect circle.
The eccentricity \( e \) is a ratio defined as \( e = \frac{c}{a} \), where \( c \) is the distance from the center to the focus of the ellipse, and \( a \) is the semi-major axis. This fraction gives us a numerical measure:

• For ellipses, \( 0 < e < 1 \) indicates that the curve is an ellipse.
• An eccentricity of 0 means the conic is a circle.
• When \( e = 1 \), the conic becomes a parabola, representing infinite elongation.
• If \( e > 1 \), we have a hyperbola, indicating two separate curves.

Understanding eccentricity helps in visualizing the shape and nature of the conic; smaller eccentricities signify more circular ellipses, while larger values approach the flattened shapes of hyperbolas.

Ellipse Identification in Conic Sections

In conic sections, identifying whether a given polar equation represents an ellipse involves examining its eccentricity. If the eccentricity is less than 1, then the conic is an ellipse. This critical information shapes our understanding of the conic's behavior and appearance.
For the equation \( r = \frac{4}{3 - 2\cos\theta} \), rewriting it to compare with the standard form \( r = \frac{ed}{1 + e\cos\theta} \): first, rearrange as \( r = \frac{4}{1 - (-\frac{2}{3})\cos\theta} \). Here, \( e = \frac{2}{3} \) since its value is less than 1, so the conic is confirmed to be an ellipse.

• Ellipses have a definite shape characterized by the elongated oval form.
• They are symmetric about their major and minor axes.
• In polar equations, an ellipse can appear stretched in various directions based on the cosine or sine factors.

Recognizing these shapes helps us graph ellipses more accurately and understand their spatial properties in the coordinate plane. With polar equations, the identification of ellipses becomes clearer and more insightful, allowing for precise mathematical and graphical interpretations.